We study theoretically spontaneous currents and magnetic field induced in a superconductor-ferromagnet (S-F) bilayer due to direct and inverse proximity effects. The induced currents {are Meissner currents that appear even in the absence of an external magnetic field due to the magnetic moment in the ferromagnet }and {to the magnetization } in the superconductor . The latter is induced by the inverse proximity effect over a distance of the order of the superconducting correlation length $xi _{S}$. On the other hand the magnetic induction $B$, caused by Meissner currents, penetrates the S film over the London length $lambda _{S}$. Even though $lambda _{S}$ usually exceeds considerably the correlation length, the amplitude and sign of $B$ at distances much larger than $xi _{S}$ depends crucially on the strength of the exchange energy in the ferromagnet and on the magnetic moment induced in the in the S layer.
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